Model 2: In this model, we take \(m_{\eta } = 32\)  GeV. We choose the interaction strength \(g_{\eta } = 1.12 \times 10^{-1}\) , we take \(g_{VB} = 1 \times 10^{-6}\) , and taking into account the constraint for \(t_{2\alpha }\)  in equation (REF ), we obtain \(s_{\alpha } \simeq -4.7\times 10^{-7}\) . Then we increase the mass of the \(Z^{\prime }\)  mediator to 3.5 GeV. The dominant interaction channel is still \(\eta + \overline{\eta } \rightarrow Z^{\prime } + Z^{\prime }\) . In this case, we find \(\langle \sigma _{ann} v\rangle \simeq 1.88 \times 10^{-26}\)  cm\(^{3}\) s\(^{-1}\)  and we fit the DM \(\Omega h^{2} \simeq 0.119\)  from the results shown in Figures REF  and REF . Our values for the spin independent cross sections of \(\eta \)  with the nuclei are \(\sigma _{SI}^{p}\simeq 6.04 \times 10^{-46}\)  and \(\sigma _{SI}^{n}\simeq 1.42\times 10^{-52}\)  cm\(^{2}\) . For \(M_{Z^{\prime }} = 3.5\)  GeV, the additional \(c\overline{c}\)  decay channel opens up, being now kinematically allowed. The \(Z^{\prime }\)  decay width, \(\Gamma _{Z^{\prime }}\) , will therefore be larger than that for Model 1, in this case \(\Gamma _{Z^{\prime }} \simeq 3.74\times 10^{-14}\)  GeV. The branching ratios for the \(Z^{\prime }\)  decay are found to be: 25% in \(u\overline{u}\) , 24.9% in \(c\overline{c}\) , 18.8% into \(e^{+}e^{-}\) , 18.8% into \(\mu ^{+}\mu ^{-}\) , 6.28% into \(d\overline{d}\) , 6.28% into \(s\overline{s}\) . Thus, in addition to the other channels of Model 1, the \(c\overline{c}\)  now also significantly contributes to \(\pi ^0 \rightarrow \gamma \gamma \)  production. For a large, \(c\overline{c}\)  decay channel it is shown in Ref. [1]} that the annihilations of DM particles of mass between \(\sim \)  28 GeV and \(\sim \)  36 GeV into \(c\overline{c}\)  channels may significantly account for the \(\sim \)  2 – 3 GeV excess.
